DIRECTED COMPACT PERCOLATION NEAR A WALL .2. CLUSTER LENGTH AND SIZE

The mean cluster size and length for the unbiassed growth of compact c lusters near a dry wall are considered. In the case of the cluster siz e below p(c) an exact expression is obtained for a seed of arbitrary w idth and distance from the surface. It is found that the critical expo nent gamma = 1 for any finite distance from the surface. Crossover to the bulk value gamma = 2 as the distance from the surface tends to inf inity is observed. This extends an existing result for the exponent be ta of the percolation probability which changes from a value of 2 in t he presence of a surface to 1 in the bulk limit. The value Delta = 3 o f the scaling size exponent is unchanged by the introduction of the su rface. The cluster size above p(c) and the mean cluster length are inv estigated using differential approximants from which we conjecture tha t these functions satisfy second-order differential equations. Accepti ng this conjecture gives a mean size exponent the same as below p(c) a nd a logarithmic divergence of the mean length from both sides of the critical point. The latter result together with scaling theory predict s that the exponent nu(parallel to) has the value 2, the same as for t he bulk problem.